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  • View in gallery
    Fig. 1.

    Schematic view of the density field near a rotating uniform slope with slope angle α for (a) geopotential and (b) rotated coordinate systems. Thin black and gray lines indicate the structure of the buoyancy b and the equilibrium buoyancy b, respectively.

  • View in gallery
    Fig. 2.

    Energy forms and pathways in the BBL. Arrows indicate typical (black) or exceptional (gray) energy conversion pathways.

  • View in gallery
    Fig. 3.

    Evolution of (a) upslope velocity, (b) upslope vertical shear, (c) buoyancy, and (d) buoyancy frequency squared for an example with downwelling-favorable flow (υ > 0). Model parameters are summarized in Table 1. Dark gray lines indicate the upper edge of the turbulent boundary layer, where ε falls below the threshold of 10−11 W kg−1; light gray lines show gravitationally unstable regions.

  • View in gallery
    Fig. 4.

    Evolution of (a) turbulence dissipation rate, (b) ratio of shear production and dissipation, (c) turbulent buoyancy flux, and (d) mixing rate of buoyancy variance for downwelling-favorable flow (υ > 0). Note the special logarithmic color scale in (c) with negative (downward) fluxes plotted in blue, and positive (upward) fluxes plotted in red. Gray lines are as in Fig. 3.

  • View in gallery
    Fig. 5.

    As in Fig. 3, but now for upwelling-favorable flow (υ < 0). Note the different depth range and color scales.

  • View in gallery
    Fig. 6.

    As in Fig. 4, but now for upwelling-favorable flow (υ < 0). Note the different depth range (color scales are identical).

  • View in gallery
    Fig. 7.

    Cumulative budgets of (a),(b) mean kinetic energy and (c),(d) mean APE of the relative BBL motions for upwelling- and downwelling favorable flow, respectively. Energy conversion terms correspond to the time-integrated versions of those in the energy budgets in (17) and (19), and are counted positive if they have a tendency to increase the respective energy compartment. Note the different scales.

  • View in gallery
    Fig. 8.

    As in Fig. 7, but now the cumulative budget of turbulence APE in (21) is shown. Note the different scales.

  • View in gallery
    Fig. 9.

    Nondimensional time for Ekman layer arrest, TE/Tf, for upwelling favorable flow as function of the slope Burger number S and (a) the roughness number R for fixed Z = 100, and (b) the frequency ratio Z for fixed R = 10−4. Selected values are plotted as white contour lines, and the slope angle α is plotted as black contour lines in (b). Note the logarithmic color scale.

  • View in gallery
    Fig. 10.

    Variability of (a),(b) the energy ratio Q and (c),(d) the bulk mixing efficiency Γ as functions of S, Z, and R. Thin contour lines indicate selected values. Only simulations to the right of the thick contour line reached full Ekman arrest (i.e., TE/Tf < 50). Note the logarithmic contour scale in (c) and (d).

  • View in gallery
    Fig. 11.

    As in Fig. 10, but for (a),(b) the nondimensional averaged dissipation rate and (c),(d) the nondimensional averaged mixing rate. Note the logarithmic contour scale in (c) and (d).

  • View in gallery
    Fig. 12.

    Variability of (a) normalized time scales for Ekman layer arrest (full line) and for the appearance of BBL oscillations; (b) ratio of arrest time scales for down- and upwelling favorable flow; (c) nondimensional maximum thicknesses of the BBL, , and the gravitationally unstable layer inside the BBL, . Note that we have chosen Z = 100 and R = 10−4 here. Gray-shaded areas indicate regions of incomplete Ekman layer arrest.

  • View in gallery
    Fig. 13.

    Variability of (a) ratio of APE to dissipated energy at Ekman arrest, (b) cumulative mixing efficiency, (c) integrated nondimensional energy dissipation, and (d) integrated nondimensional mixing rate for up- and downwelling-favorable flow. The meaning of the different line types is explained in (a). Gray-shaded areas, and values of Z and R, are as in Fig. 12.

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Energetics of Bottom Ekman Layers during Buoyancy Arrest

Lars UmlaufLeibniz-Institute for Baltic Sea Research, Warnemünde, Germany

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William D. SmythOregon State University, Corvallis, Oregon

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James N. MoumOregon State University, Corvallis, Oregon

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Abstract

Turbulent bottom Ekman layers are among the most important energy conversion sites in the ocean. Their energetics are notoriously complex, in particular near sloping topography, where the feedback between cross-slope Ekman transports, buoyancy forcing, and mixing affects the energy budget in ways that are not well understood. Here, the authors attempt to clarify the energy pathways and different routes to mixing, using a combined theoretical and modeling approach. The analysis is based on a newly developed energy flux diagram for turbulent Ekman layers near sloping topography that allows for an exact definition of the different energy reservoirs and energy pathways. Using a second-moment turbulence model, it is shown that mixing efficiencies increase for increasing slope angle and interior stratification, but do not exceed the threshold of 5% except for very steep slopes, where the canonical value of 20% may be reached. Available potential energy generated by cross-slope advection may equal up to 70% of the energy lost to dissipation for upwelling-favorable flow, and up to 40% for downwelling-favorable flow.

Corresponding author address: Lars Umlauf, Leibniz-Institute for Baltic Sea Research, Seestrasse 15, 18119 Warnemünde, Germany. E-mail: lars.umlauf@io-warnemuende.de

Abstract

Turbulent bottom Ekman layers are among the most important energy conversion sites in the ocean. Their energetics are notoriously complex, in particular near sloping topography, where the feedback between cross-slope Ekman transports, buoyancy forcing, and mixing affects the energy budget in ways that are not well understood. Here, the authors attempt to clarify the energy pathways and different routes to mixing, using a combined theoretical and modeling approach. The analysis is based on a newly developed energy flux diagram for turbulent Ekman layers near sloping topography that allows for an exact definition of the different energy reservoirs and energy pathways. Using a second-moment turbulence model, it is shown that mixing efficiencies increase for increasing slope angle and interior stratification, but do not exceed the threshold of 5% except for very steep slopes, where the canonical value of 20% may be reached. Available potential energy generated by cross-slope advection may equal up to 70% of the energy lost to dissipation for upwelling-favorable flow, and up to 40% for downwelling-favorable flow.

Corresponding author address: Lars Umlauf, Leibniz-Institute for Baltic Sea Research, Seestrasse 15, 18119 Warnemünde, Germany. E-mail: lars.umlauf@io-warnemuende.de

1. Introduction

A stratified, geostrophically balanced current flowing along sloping topography is known to induce up- or downslope Ekman transports inside a turbulent near-bottom region that is usually referred to as the bottom Ekman layer. The currents associated with these Ekman transports have a number of important dynamical implications due to their ability to set up baroclinic pressure gradients by cross-slope advection of near-bottom isopycnals.

Previous investigations of this effect have shown that, after an initial adjustment phase, buoyancy effects may become strong enough to stop the cross-slope Ekman transport (MacCready and Rhines 1991; Trowbridge and Lentz 1991). Such “arrested” Ekman layers are in thermal-wind balance, and satisfy the no-slip boundary condition at the bottom without supporting any turbulent or viscous stress—evidently with significant implications for the spindown of interior flows (MacCready and Rhines 1993; Garrett et al. 1993). More recent studies of this process have therefore focused on the description of the vertical structure of the Ekman layer in the final arrested stage in order to arrive at refined expressions for the shutdown time scales and final bottom boundary layer (BBL) thicknesses (e.g., Middleton and Ramsden 1996; Brink and Lentz 2010, hereinafter BL10). The basic physical mechanisms proposed in these theoretical investigations are largely supported by field observations (e.g., Trowbridge and Lentz 1998; Perlin et al. 2005a).

Despite a wealth of studies focusing on buoyancy arrest, the energy pathways and different routes to mixing and dissipation remain largely unexplored. The major difficulty lies in the quantification of the different energy reservoirs and energy conversion terms, which is not straightforward in view of the complex vertical structure of these near-bottom flows affected by stratification, rotation, and the proximity of a sloping bottom (Perlin et al. 2005b, 2007). An additional complication has been highlighted by Moum et al. (2004), who pointed out that in situations with downslope Ekman transport, light water may be advected underneath dense water, leading to turbulent convection. This process provides an additional energy source for turbulence (besides bottom friction) with yet unknown implications for the overall energetics of the BBL. A related effect has been observed in lakes, where differential advection due to friction-induced shear has been shown to create gravitationally unstable layers (Lorke et al. 2008; Umlauf and Burchard 2011; Becherer and Umlauf 2011).

From a practical point of view, it would be desirable to better constrain the mixing efficiency in complex turbulent BBLs. There is growing evidence that in the near-bottom region, or generally in energetic turbulent flows, the mixing efficiency may be substantially smaller than the canonical value of 0.2 (e.g., Bluteau et al. 2013). The mixing efficiency is needed to convert observed dissipation rates (e.g., from turbulence microstructure measurements) into buoyancy fluxes and turbulent diffusivities, which govern the near-bottom transport of dissolved substances and suspended particulate matter.

In the following, we investigate this problem with a combined theoretical and numerical modeling approach, based on the idealized geometry described in the following section. In section 3, we develop a theoretical framework describing the different energy reservoirs and energy pathways for buoyancy-controlled Ekman layers, and in section 4, we identify the nondimensional parameters that determine the problem. In section 5, the energetics of a few idealized examples are investigated before we explore the entire physically relevant parameter space in section 6.

2. Geometry and model equations

a. Geometry

We consider the motion of a stratified Boussinesq fluid in the vicinity of a uniform slope with constant slope angle α, rotating about the vertical axes at the constant rate f/2 (Fig. 1a). The buoyancy frequency for this geometry is defined as
e1
where is the vertical coordinate (pointing upward), and
e2
denotes buoyancy. Here, ρ is density, ρ0 is a constant reference density, and g is the gravitational acceleration. All quantities are Reynolds averaged unless noted otherwise. Deviations from the Reynolds average are denoted by a prime (as in b′), and assumed to be due to turbulent motions. Random internal-wave motions are not considered.
Fig. 1.
Fig. 1.

Schematic view of the density field near a rotating uniform slope with slope angle α for (a) geopotential and (b) rotated coordinate systems. Thin black and gray lines indicate the structure of the buoyancy b and the equilibrium buoyancy b, respectively.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0041.1

Far away from the bottom, the buoyancy frequency is assumed to approach the constant value N. Isopycnals in this interior region remain strictly horizontal at all times but may exhibit vertical displacements due to up- and downslope advection. The buoyancy thus converges to the background or equilibrium buoyancy b, which is, in general, a function of and time.

Following previous authors (e.g., Umlauf and Burchard 2011; Garrett et al. 1993), we assume that all gradients, except the buoyancy gradient, vanish in the upslope direction. By introducing a rotated coordinate system with the x axis pointing upslope and the z axis directed normal to the slope (Fig. 1), the problem is made one-dimensional in z. Although restrictive, the assumption of upslope homogeneity greatly simplifies the theoretical analysis, and has as a result been adopted in numerous previous studies (Phillips 1970; Thorpe 1987; Garrett 1990).

Under these assumptions, the upslope buoyancy gradient is constant, and related to the interior stratification according to
e3
The slope-normal buoyancy gradient, defined as
e4
evolves due to the combined effects of near-bottom mixing and shear (Umlauf and Burchard 2011). It can be shown that and the vertical buoyancy gradient N2 are related according to
e5

b. Model equations

For the above geometry and modeling assumptions, it is straightforward to show (see, e.g., Garrett et al. 1993; Umlauf and Burchard 2011) that the Reynolds-averaged Boussinesq equations and the transport equation for buoyancy can be written as
e6
where u and υ denote the upslope and slope-normal velocities, respectively. In the momentum equations in (6), we introduced the integration constants Px(t) and Py(t), which play the role of prescribed pressure gradients. In the buoyancy equation, we used (3) to express the upslope buoyancy gradient in terms of N. For convenience, we introduced the abbreviations and for the Reynolds stresses (normalized by ρ0), and for the slope-normal turbulent buoyancy flux (the overbar denotes the Reynolds average here). Assuming high Reynolds number, molecular fluxes are negligible in (6). The turbulent fluxes were computed from a second-moment turbulence model that is described in the appendix.
The first term on the right-hand side of the u equation in (6) involves the difference between the actual buoyancy b and the equilibrium buoyancy b and represents the tendency of isopycnals to relax back to their equilibrium positions. An equation for b is easily derived by evaluating the full buoyancy equation in (6) for z → ∞, yielding
e7
where we have introduced u(t) to denote the spatially constant upslope velocity far away from the bottom. The initial buoyancy profile is obtained by integrating (1) under the assumption that isopycnals are initially horizontal everywhere:
e8
The equations of motion in (6) are solved using the bottom boundary conditions
e9
We further assume that the region above the bottom boundary layer (z → ∞) is nonturbulent, implying that all turbulent fluxes vanish.

c. Turbulence kinetic energy

A central component of both the BBL energy budget discussed below and the turbulence model described in the appendix is the transport equation for the turbulence kinetic energy (TKE):
e10
where Fk is the vertical turbulent flux of TKE. The quantities P and B appearing in (10) denote the turbulence shear and buoyancy production:
e11
where B is recognized as the vertical component of the total turbulent buoyancy flux. The dissipation rate ε is computed from a transport equation of structure similar to (10) as described in the appendix.

d. Molecular mixing rate

Irreversible mixing is quantified using the molecular destruction rate of buoyancy variance:
e12
where νb is the molecular diffusivity of buoyancy, corresponding to that of either heat or salt (for simplicity, only a single stratifying component is investigated here). On the right-hand side of (12), summation over i = 1, 2, 3 is implied with the xi denoting some arbitrary but orthogonal coordinates. The quantity defined in (12) is positive by definition, and describes the irreversible mixing of fluid with different densities that, as shown below, is directly related to the irreversible loss of available potential energy.
The mixing rate χb appears in a well-known transport equation for the buoyancy variance (Tennekes and Lumley 1972), which, using the geometric assumptions introduced in section 2, can be written as
e13
where Fb denotes the sum of the turbulent and viscous transport terms that vanishes at z = 0, ∞ (this property will be used below). The production term is given by
e14
using (3) and (4), see Umlauf and Burchard (2005). Note that (14) involves the unknown upslope buoyancy flux that is determined by processes not described by our turbulence model. For the small slope angles considered in the following, it is likely (but cannot be proven) that the contribution to Pb due to this flux is negligible.

3. Energetics

The energetics of the BBL are most conveniently studied by focusing on the near-bottom motions relative to the prescribed interior velocities u and υ. The corresponding equations for the relative velocities, and , and the buoyancy anomaly, , are derived by evaluating (6) at z → ∞ and subtracting the result from the original equation:
e15
Compared to (6), the equations in (15) have the practical advantage that slope-normal integrals across our semi-infinite domain remain finite.

a. Kinetic energy

An equation for the kinetic energy contained in the relative motions of the boundary layer:
e16
can be obtained by multiplying the first two equations in (15) with and , respectively, and integrating their sum across the BBL. This yields the following:
e17
illustrating that changes in kinetic energy are due to work performed against gravity and pressure forces (first term under the integral on the right-hand side), conversion into turbulent kinetic energy via shear production (second term), and work performed by the background flow against the bottom stresses and (last two terms). The bottom stress typically opposes the interior flow such that the latter terms provide a source of kinetic energy for the relative motions. The work represented by the first term (work against gravity and pressure forces) is converted into a special form of potential energy as discussed in the following.

b. Potential energy

Available potential energy (APE) is generally understood as the amount of energy released when a stratified fluid is adiabatically redistributed into a state of minimum (or background) potential energy (Winters et al. 1995). For the special geometry considered here, APE can be conveniently quantified by investigating the energy released when all fluid particles in the BBL adiabatically relax back (up- or downslope) to their equilibrium positions. Evidently, this amount of energy is equal to the work performed by adiabatically displacing fluid particles from their equilibrium positions into the actual density configuration, which forms the starting point of the following analysis.

The difference between the local background buoyancy, b, and the buoyancy of a fluid particle displaced from its equilibrium position by the distance δ in the upslope direction is . Using (3), this can also be written as . In an Eulerian framework, the particle’s buoyancy coincides with the local buoyancy, b, from which we conclude that Δb and describe the same quantity. From the first term on the right-hand side of the u-momentum equation in (6) it is therefore evident that a force (per unit mass) of magnitude acts on the particle. Integrating this result over the particle path, it is easy to show that the work performed against gravity and pressure forces (i.e., the APE associated with the particle) is equal to .

Using the relation derived above, this amount of energy can also be written as , independent of the particle path and slope angle. The APE of the BBL is therefore
e18
for which a transport equation can be derived by multiplying the buoyancy equation in (15) by the factor . After some rearrangements, this yields an expression of the following form:
e19
where we have used the fact that the slope-normal buoyancy flux, G, vanishes at the integration limits, and introduced the energy conversion term:
e20
The first integral on the right-hand side of (19) is seen to appear with opposite sign in (17), and thus describes the conversion between mean kinetic energy and mean APE. The meaning of becomes clear after multiplying the transport equation [(13)] by the factor , and integrating the result in the slope-normal direction:
e21
Here, we use (11), and take into account that the flux of buoyancy variance Fb vanishes at both integration limits. The integral on the left-hand side of (21) describes the APE associated with the turbulent buoyancy fluctuations, which will be referred to as the “turbulence” APE. This term is seen to change as a result of the three source and sink terms on the right-hand side of (21). The first term describes the conversion from or to TKE via the vertical buoyancy flux B that appears with the opposite sign in (10), whereas the last term represents the irreversible loss due to mixing of small-scale density gradients. The second term, , appears with the opposite sign in (19), and thus reflects the conversion between the mean APE and turbulence APE.

c. Energy pathways

For the physical interpretation of the above relations it is helpful to revisit a few aspects of the work by Winters et al. (1995), who investigated the energetics of stably stratified Boussinesq fluids. Their analysis was based on the adiabatic resorting (or restratification) of all fluid particles inside a given volume into a state of minimum potential energy (i.e., with the densest fluid at the bottom). If we denote the vertical position of a fluid particle after sorting by , and the corresponding vertical buoyancy profile by , then defines the (strictly positive) buoyancy frequency in the sorted state. As one of their main results, Winters et al. (1995) showed that the instantaneous gain of background potential energy due to mixing is
e22
where, as in (12), we have introduced the arbitrary orthogonal coordinates xi, and sum over i = 1, 2, 3. Note that the buoyancy gradients in (22) are based on turbulent fluctuations, whereas Winters et al. (1995) used the full instantaneous buoyancy fields. For the high Reynolds number flows considered here, however, the difference is negligible.
Because the sorting volume is infinite for our geometry while changes in potential energy in the BBL remain finite, the sorted buoyancy field will be indistinguishable from the equilibrium buoyancy b. Further, since the sorted and equilibrium buoyancy fields are identical, the sorted buoyancy gradient in (22) will coincide with the constant background buoyancy gradient . The Reynolds-averaged version of (22) for our geometry, therefore, is the following:
e23
where we have replaced the volume integral in (22) by slope-normal integration in view of the lateral homogeneity of our problem. Using the definition of χb in (12), the quantity defined in (23) is seen to be identical to the last term in (21), illustrating that, for our geometry, the gain of background potential energy is balanced by a loss of (turbulence) APE. Following Winters et al. (1995), we will therefore use as a measure for “mixing” in the following.
We next add the budgets for the turbulence and mean kinetic energies, in (10) and (17), to the potential energy budgets in (19) and (21). All energy conversion terms cancel, and the following equation for the total energy of the relative BBL motions is derived:
e24
where we have introduced the integrated dissipation rate:
e25
The total energy budget in (24) shows that the relative BBL motions gain mechanical energy from the background flow via turbulent bottom friction (first two terms on the right-hand side), while irreversibly losing energy by molecular mixing of small-scale density gradients (conversion to background potential energy, see above) and viscous dissipation (conversion to internal energy). The ratio of the two energy dissipation rates:
e26
defines the mixing efficiency, which is of central importance for the description of mixing in stably stratified flows (Winters et al. 1995).

The different energy forms and pathways discussed above are schematically depicted in Fig. 2. It is worth noting that Scotti and White (2014) recently derived a similar energy flux diagram from a more general context. Those authors analyzed mixing in turbulent flows using a locally defined APE that, for the special geometry used here, coincides with our expression derived in section 3b. In their Fig. 4, Scotti and White (2014) present an energy diagram for small displacements from the sorted state that, if integrated in the slope-normal direction, leads to exactly the same four energy reservoirs as those shown in our Fig. 2. As pointed out by Scotti and White (2014), their results for small displacements become exact if the stratification in the sorted state is uniform. In our case, = N = const (see above), which corroborates the findings above. Finally, it should be noted that the energy budgets expressed by the relations in (17)(24) are exact (i.e., they do not involve any turbulence modeling assumptions except that of high Reynolds number).

Fig. 2.
Fig. 2.

Energy forms and pathways in the BBL. Arrows indicate typical (black) or exceptional (gray) energy conversion pathways.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0041.1

d. Boundary layer resonance

The conversion between kinetic and potential energy through the buoyancy flux terms in (17) and (19) implies the possibility of reversible oscillations. To investigate this, we take the time derivative of the x component of the momentum budget in (15), and insert the second and third relations in (15), arriving at the following:
e27
with . Equation (27) describes a forced harmonic oscillator with critical (resonance) frequency ωc, where the obvious limiting cases are ωc = f for α = 0 (horizontal inertial oscillations), and ωc = N for α = π/2 (vertical oscillations at the buoyancy frequency). Interestingly, for all intermediate cases with 0 < α < π/2, the resonance frequency ωc exactly corresponds to the frequency for the critical reflection of internal waves impinging on a slope with slope angle α (Thorpe 2005). Note, however, that solutions of (27) in the inviscid, adiabatic limit (right-hand side vanishes) are purely oscillatory rather than wavelike (e.g., there is no dispersion relation and thus no phase propagation in this limit). In view of this, these one-dimensional solutions are likely to be relevant only if the lateral wavelength is much larger than the thickness of the BBL.

4. Nondimensional relations

Following previous studies on buoyancy-controlled bottom Ekman layers (e.g., MacCready and Rhines 1993; Middleton and Ramsden 1996; BL10), we assume from now on that the interior flow (outside the BBL) is determined by a stationary, geostrophically balanced along-slope current, abruptly starting from rest at t = 0. With these specifications, the problem is determined by five parameters: the bottom slope α, the rotation rate f, the interior stratification N, the magnitude of the interior flow V = |υ|, and the bottom roughness z0. These parameters may be used to define the nondimensional time and bottom distance via the expressions and , as well as a set of nondimensional variables of the following form:
e28
The equations of motion in (6) can then be rewritten in nondimensional form:
e29
where we have again assumed that the slope angle is small (α ≪ 1). From (29), the slope Burger number S = αNf−1 and the frequency ratio Z = Nf−1 are identified as the key nondimensional parameters of the problem. A third nondimensional quantity, R = z0NV−1, referred to as the roughness number in the following, arises from the nondimensional form of the lower boundary condition for the dissipation rate introduced in the context of (A7) in the appendix. The quantity R may be viewed as a nondimensional measure of the bottom roughness.

Whereas the slope Burger number S is a familiar parameter in studies of Ekman layer arrest, the effect of the bottom roughness is often described in a different way by introducing the nondimensional parameter d = CdZ, where Cd is the quadratic drag coefficient (Trowbridge and Lentz 1991; MacCready and Rhines 1993; Ramsden 1995; BL10). This approach, however, requires matching the law of the wall at some small, but otherwise arbitrary distance from the bottom (see BL10), which introduces an additional length scale to the problem that cannot be uniquely determined from physical arguments. The use of the parameter R avoids this problem. In this context, it should be noted that the parameter Z could be rescaled, without loss of generality, by some constant factor of order 10−3 (a typical value for the drag coefficient) to yield stress terms of order 1 in (29), and Z ~ d.

In summary, the problem studied here is mathematically determined by the three parameters S, Z, and R that may be interpreted as nondimensional measures of the slope, the interior stratification, and the bottom roughness, respectively. As these parameters are found from purely dimensional arguments, they are expected to appear in any theoretical or numerical model analysis.

5. Forcing with constant alongslope flow

In the following, we investigate the properties of the system in (6) with the help of some examples for upwelling- and downwelling-favorable flow for some typical parameters summarized in Table 1. The geostrophically balanced interior velocity is directed along the slope (see previous section) and has an absolute value of V = 0.2 m s−1. The simulation period in these examples is 20 days, which is sufficient to reveal all physically relevant effects.

Table 1.

Parameters used for the simulations in section 5.

Table 1.

a. Downwelling-favorable flow

In our first example, the interior flow is downwelling favorable (υ > 0), thus inducing a downslope Ekman transport inside the BBL. Figure 3a illustrates that the BBL (here defined as the near-bottom region where the dissipation rate exceeds the minimum threshold of ε = 10−11 W kg−1) reaches a thickness of approximately 90 m with no indication of an asymptotic state. The theoretical BBL thickness in the fully arrested state, computed from the model of BL10, is 207 m, reached after an adjustment time of 95 days. The latter is sensitive with respect to the parameterization of bottom friction, and we have therefore adjusted the parameters used by BL10 for the somewhat larger bottom roughness used in our study. In view of the natural variability of such systems, and a well-known instability that is known to grow in more realistic two-dimensional simulations (Allen and Newberger 1996), these numbers indicate that Ekman layer arrest is unlikely to play a role in this example.

Fig. 3.
Fig. 3.

Evolution of (a) upslope velocity, (b) upslope vertical shear, (c) buoyancy, and (d) buoyancy frequency squared for an example with downwelling-favorable flow (υ > 0). Model parameters are summarized in Table 1. Dark gray lines indicate the upper edge of the turbulent boundary layer, where ε falls below the threshold of 10−11 W kg−1; light gray lines show gravitationally unstable regions.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0041.1

Before analyzing the results from this example, we want to emphasize that this and many other downwelling-favorable simulations show an oscillatory instability for longer simulation periods that grows until some saturation amplitude seems to be reached (see Fig. 3 for t > 350 h). Here, these oscillations exhibit a period of approximately 17 h, which is close to the natural BBL resonance period Tc = 2π/ωc = 17.03 h (see section 3d). Extensive tests have shown that numerical reasons for these oscillations can be excluded. Similar oscillations have also been found in other studies of this type (see, e.g., Fig. 1 in BL10), but the nature of the instability mechanism triggering them as well as their physical significance is so far unexplored. We will therefore focus most of our discussion of the downwelling-favorable cases to the initial period, before the oscillations become evident.

The downslope velocities associated with the Ekman transport for the case studied here are seen to be of the order of a few centimeters per second (Fig. 3a), resulting in a continuous downslope transport of buoyant fluid and a corresponding increase of the local buoyancy (Fig. 3c). The most remarkable feature evident from the stratification shown in Fig. 3d is the generation of a gravitationally unstable near-bottom layer that reaches a thickness of approximately 30 m in this example. The generation of such unstable near-bottom regions can be understood from the evolution equation for the squared buoyancy frequency:
e30
which may be derived from the time derivative of (5), using the z derivative of the buoyancy equation in (6), and introducing the upslope shear Su ≡ ∂u/∂z. The expression in (30) illustrates the competing effects of shear and mixing for the generation or destruction of stratification inside the BBL. Specifically, if mixing is not too strong, gravitationally unstable layers (N2 < 0) may be generated in regions with positive shear (Su > 0), noting that sinα > 0 in all our simulations. Figure 3b shows that Su > 0 is indeed observed in the lower part of the BBL as a result of the near-bottom intensification of the Ekman transport visible in Fig. 3a. In these regions, light water from upslope locations is advected underneath dense water thus leading to a gravitational destabilization of the BBL. This is exactly the differential advection mechanism suggested by Moum et al. (2004) from their observations of convectively driven mixing in the bottom boundary layer on the Oregon shelf.

The impact of these unstable layers on turbulence and mixing in the BBL is illustrated in Fig. 4. The dissipation rate (Fig. 4a) strongly increases close to the bottom, with a 1/z dependency that is consistent with the law-of-the-wall scaling in (A7). This is in agreement with the fact that shear production and dissipation are approximately in balance in this region (Fig. 4b), and shows, surprisingly, that the contribution of the buoyancy flux to the total TKE budget is negligible inside the convective layer (tests with some alternative turbulence models described in the appendix support this finding). The most pronounced impact of the unstable layers, however, becomes evident from the distribution of the vertical turbulent buoyancy flux shown in Fig. 4c. While the buoyancy flux in the upper part of the BBL is negative (i.e., downward, as expected in a stably stratified fluid), the unstably stratified near-bottom region exhibits a positive (upward) flux of comparable magnitude. Direct observations of the turbulent buoyancy flux in lakes are consistent with this finding (Lorke et al. 2008). The most important conclusion from this is that the classical approach of inferring the turbulent buoyancy flux B from observed dissipation rates (e.g., obtained from microstructure measurements) via the relation B = −γε (where γ ≈ 0.2) breaks down in boundary layers of this type. Comparing Figs. 4a and 4c, it is evident that γ, in the context of microstructure measurements often assumed to be a constant and positive “mixing coefficient,” is neither constant nor positive (see related discussion in Gargett and Moum 1995).

Fig. 4.
Fig. 4.

Evolution of (a) turbulence dissipation rate, (b) ratio of shear production and dissipation, (c) turbulent buoyancy flux, and (d) mixing rate of buoyancy variance for downwelling-favorable flow (υ > 0). Note the special logarithmic color scale in (c) with negative (downward) fluxes plotted in blue, and positive (upward) fluxes plotted in red. Gray lines are as in Fig. 3.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0041.1

As outlined above, energetically consistent definitions of mixing and mixing efficiency involve the integral of the molecular mixing rate χb. The spatial distribution of this quantity, shown in Fig. 4d, illustrates that the contribution of the convective layers to overall mixing inside the BBL is negligible, despite the large buoyancy fluxes (Fig. 4c) and large turbulent diffusivities (not shown) found in these regions. Most of the mixing occurs in the entrainment layer, where stratification is strong and mixing, partly fueled by the energy diffused out of the convective layers, is efficient.

b. Upwelling-favorable flow

The dynamics of the upwelling-favorable case (υ < 0) has been investigated in detail by BL10, and here we only briefly review their main results in the context of our simulations. These authors pointed out that for small slope Burger numbers (here S = 0.22), a permanent density interface evolves at the top of the BBL after a rapid initial entrainment phase that is essentially completed within one natural oscillation period Tc.

This behavior is mirrored in Fig. 5, which shows the evolution of an upwelling-favorable BBL with parameters equal to those used for the downwelling case except for the sign of υ (Table 1).

Fig. 5.
Fig. 5.

As in Fig. 3, but now for upwelling-favorable flow (υ < 0). Note the different depth range and color scales.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0041.1

Striking differences from the downwelling case are the much smaller BBL thickness and the collapse of entrainment after only one day. The model of BL10 predicts 15.3 m for the final BBL thickness, slightly smaller than the value of 18 m found in our simulations, which may be due to the uncertainty in the drag coefficient appearing in their expressions (see discussion above). In contrast to the downwelling-favorable case, the onset of Ekman layer arrest is clearly evident from the steady decrease of both the Ekman-induced upslope velocity and the thickness of the turbulent near-bottom layer (Fig. 5a). By the end of the simulation, the bottom stress (not shown) has decayed to approximately 10% of its maximum value, indicating that Ekman layer arrest is nearly complete. BL10 predict a shutdown time scale of 90 h for our parameters, which seems to be substantially shorter than the value suggested by our simulations. Their Fig. 5, however, shows that complete arrest requires at least 5–10 shutdown time scales, which again is perfectly consistent with our data.

Another indication for the shutdown process is the appearance of a strongly stratified, nonturbulent region that propagates down from the density interface toward the bottom during the adjustment process (Fig. 5d). This layer exhibits weakly damped oscillations at the resonance frequency ωc, as most clearly evident from the slope-normal shear shown in Fig. 5b. If averaged over one oscillation period, this upper part of the BBL is in thermal wind balance with the along-slope velocity (not shown), and thus characterizes the arrested BBL after full restratification (BL10).

Unstable stratification is observed only in a thin (thickness < 1 m) near-bottom layer, where the positive (destabilizing) near-bottom shear induced by bottom friction (Fig. 5b) is strong enough to overcome the stabilizing effect of mixing. In contrast to the Ekman-driven destabilization discussed above, this process does not require rotational effects. For example, it has been shown to play a significant role in lakes, where oscillating near-bottom shear induced by long internal waves (“internal seiches”) may periodically generate unstable near-bottom layers of a few meters thickness (Lorke et al. 2008; Umlauf and Burchard 2011; Becherer and Umlauf 2011). In our example, where rotational effects are crucial, these layers are less prominent and much thinner due to the strong along-slope current that is absent in the nonrotating case. According to (30), the additional turbulence associated with this current suppresses shear-induced convection by mixing stratification down from the stably stratified upper part of the BBL.

The turbulent quantities shown in Fig. 6 reveal a strongly turbulent BBL with turbulence in equilibrium nearly everywhere. The steadily growing restratification layer below the density cap (see Fig. 5d) is essentially nonturbulent, despite the presence of relatively strong shear associated with the resonant oscillations. While energy levels (Fig. 6a) and turbulent buoyancy fluxes (Fig. 6c) are comparable in magnitude to the downwelling-favorable case shown in Fig. 4, mixing in the upwelling-favorable case is about an order of magnitude larger (Fig. 6d). This reveals a higher efficiency of mixing due to the stronger shear-driven restratification in this case, and may indicate that, although the BBL is much thinner compared to the downwelling case, net mixing may be larger. This question will be investigated in the following.

Fig. 6.
Fig. 6.

As in Fig. 4, but now for upwelling-favorable flow (υ < 0). Note the different depth range (color scales are identical).

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0041.1

c. Energy pathways

While the dynamics of upwelling- and downwelling-favorable flows have been investigated in detail in previous studies, the energy pathways and different routes to mixing are not well understood. In the following, we investigate this point more closely, using the energetic framework developed above. We will work with cumulative time integrals of the different energy conversion terms to make them directly comparable to the energies stored in the four reservoirs displayed in Fig. 2 (we use the convention that conversion terms are counted positive if they have a tendency to increase energy inside a reservoir). Taking the budget in (17) for the mean kinetic energy Ek as an example, the cumulative work of the bottom stress is then expressed as , and the cumulative gain of Ek at the expense of mean APE as . All energies and conversion terms are multiplied by ρ0 to yield the “work done per unit area” (with units of Joules per square meter).

Figures 7a,b illustrate that the budget for Ek expressed by (17) is, to first order, dominated by a balance between the gain of kinetic energy from the work of the bottom stress, and the loss to turbulence due to shear production. As the energy stored in Ek is negligible (less than 1% in this example), the residuals in Figs. 7a,b represent the conversion of mean kinetic energy into mean APE (see Fig. 2). This conversion processes is significant in both cases, accounting for 15% (downwelling) and 43% (upwelling) of the cumulative energy loss due to dissipation at the end of the simulations. The unexpectedly rapid conversion rate in the upwelling-favorable case indicates that this type of boundary layer flows is surprisingly efficient in converting kinetic into potential energy. We will return to this point below.

Fig. 7.
Fig. 7.

Cumulative budgets of (a),(b) mean kinetic energy and (c),(d) mean APE of the relative BBL motions for upwelling- and downwelling favorable flow, respectively. Energy conversion terms correspond to the time-integrated versions of those in the energy budgets in (17) and (19), and are counted positive if they have a tendency to increase the respective energy compartment. Note the different scales.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0041.1

Figures 7c,d focus on the budget of mean APE in (19), showing that the observed increase in Ep can be explained almost entirely by the conversion of mean kinetic energy (as noted above). Although Fig. 2 also suggests the possibility of modifying the mean APE by turbulent stirring (term denoted by ), we find that this pathway becomes relevant only during the first few hours of our simulations, when entrainment and mixing have a stronger impact on the density structure than up- and downslope advection of isopycnals. It is worth noting that the total amount of APE is comparable in both cases, despite the fact that the BBL in the upwelling-favorable case is approximately 5 times thinner and exhibits a different density structure compared to the downwelling case.

While the stirring term is insignificant for the budget of mean APE, it becomes important in the budget of the turbulence APE in (21). Focusing first on the more straightforward upwelling-favorable case (Fig. 8b), we see that this term is negligible only during the first 120 h (5 days), characterized by gradual BBL growth due to enhanced mixing at the upper edge of the BBL (Fig. 6d). Figure 8b shows that the energy required for this entrainment process is provided almost exclusively by a conversion of TKE via the turbulent buoyancy flux (changes in turbulence APE are negligible), which is the classical picture for the energy pathway in a stably stratified turbulent flow. However, the situation changes completely after approximately 120 h, when Ekman layer arrest and BBL restratification become increasingly important. Figure 8b illustrates that during this phase, while a conversion of TKE into turbulence APE is still observed, the largest part of the energy gained from TKE is converted into mean APE (stirring term ), and therefore does not become available for mixing. Mixing rates collapse and the common interpretation of the vertical buoyancy flux as a measure of mixing breaks down.

Fig. 8.
Fig. 8.

As in Fig. 7, but now the cumulative budget of turbulence APE in (21) is shown. Note the different scales.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0041.1

The downwelling-favorable case shown in Fig. 8a exhibits additional complexity due to the appearance of convective layers. This case is comparable to the upwelling case only during the first 1–2 days, when both mixing and the transformation of turbulence APE into mean APE draw energy from the TKE. After approximately 35 h, however, the reversal of the buoyancy flux inside the convective layers (see Fig. 4c) becomes a dominant effect that induces a reversal of these energy pathways. From this point on, TKE feeds on turbulence APE, which in turn gains energy from mean APE due to the presence of gravitationally unstable layers (Fig. 8a).

The gradual onset of BBL restratification results in a second reversal of the energy pathways after approximately 150 h, when the energy conversion terms recover their original signs. During this last phase of the simulation, both the energy gain from TKE and the loss to mean APE grow quickly, however, with no evident effect on mixing. As the physical significance of the late-stage results for the downwelling-favorable case is questionable due to the appearance of the BBL oscillations described above (see Fig. 4), we do not focus on that period here. A robust result, however, is that the net mixing rate is more than 5 times smaller compared to the upwelling situation, despite the much thicker BBL and the significantly higher energy dissipation rates during downwelling.

6. Results in nondimensional parameter space

We showed in section 4 that solutions of the nondimensional equations in (29) are determined by three nondimensional parameters: the slope Burger number, S = αNf−1; the frequency ratio, Z = Nf−1; and the roughness number, R = z0NV−1. In this section, we describe an extensive suite of computations that span the oceanographically relevant region of the {S, Z, R} parameter space. Numerical computations were performed in dimensional space, and the results were then made dimensionless with the help of the external physical parameters determined in section 4. As different realizations in dimensional space have to map onto a single nondimensional representation for identical S, Z, and R, this approach provides an additional test for the correctness of the mathematical and numerical implementation of the model.

a. Upwelling-favorable flow

We start with the more straightforward case of upwelling-favorable flow (υ < 0), where our parameter space covers slope Burger numbers in the range 0.1 ≤ S ≤ 2 (cf. previous studies, see BL10), and frequency ratios of 10 ≤ Z ≤ 1000. Because α = S/Z, our analysis spans slope angles in the range 10−4α ≤ 0.2, which includes virtually all physically relevant slopes. The roughness number covers the range 10−5R ≤ 10−3, which would correspond to bottom roughnesses in the range 10−4z0 ≤ 10−2 m for some typical values of speed (e.g., V = 0.1 m s−1) and stratification (e.g., N = 10−2 s−1). Each dimension in this parameter space is sampled with 50 logarithmically spaced points, resulting in 2500 simulations for each of the two-dimensional SR or SZ slices of the parameter space discussed in the following.

To condense the wealth of data obtained from these simulations, we integrated (or averaged) selected quantities over the entire “active” period of the BBL (i.e., the period from the start of the simulations until full Ekman layer arrest has occurred). As an indicator for the latter, we chose a lower threshold for the nondimensional along-slope bottom stress, , which can be interpreted as a quadratic bottom drag coefficient. This drag coefficient decays from typical values of the order of initially toward smaller and smaller values, indicating that the Ekman layer becomes increasingly “slippery” during arrest (MacCready and Rhines 1993). If the bottom drag coefficient falls below a threshold of , approximately two orders of magnitude below the typical values for a BBL over flat bottom, we define the Ekman layer as “fully arrested.”

Clearly, this threshold is somewhat arbitrary but all of the following results turned out to be insensitive with respect to small variations of it. The time TE required for full Ekman layer arrest, however, obviously does depend on the exact value of this threshold. This should be kept in mind when comparing our estimates of the arrest time with those found in other studies (e.g., Middleton and Ramsden 1996; BL10). Note that all simulations were stopped after a maximum of 50 inertial periods (t/Tf = 50), implying that a small subset of our simulations (usually cases with extremely gentle slope) did not reach full arrest. Those cases will be identified where appropriate.

Figure 9a shows the arrest time TE, made dimensionless with the inertial period Tf, as a function of the slope Burger number S and roughness number R, choosing the intermediate value Z = Nf−1 = 100 for the frequency ratio. While there is only a small dependence on R, variations in S are seen to result in a significant variability of the arrest time scale. For large S (steep slopes, strong stratification), Ekman layer arrest may occur within less than one inertial period. Figure 9b, showing TE/Tf as a function of S and Z for an intermediate roughness number (R = 10−4), illustrates that stronger stratification (increasing Z) generally results in faster Ekman layer arrest. However, the variation is smaller than the variation with S over the oceanographic range. Particularly interesting is the unexpected increase of the arrest time scale for very small Z and large S visible in Fig. 9b. This parameter combination represents the situation on very steep slopes (α ≈ 0.1, see Fig. 9b), and will be discussed in more detail below.

Fig. 9.
Fig. 9.

Nondimensional time for Ekman layer arrest, TE/Tf, for upwelling favorable flow as function of the slope Burger number S and (a) the roughness number R for fixed Z = 100, and (b) the frequency ratio Z for fixed R = 10−4. Selected values are plotted as white contour lines, and the slope angle α is plotted as black contour lines in (b). Note the logarithmic color scale.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0041.1

Available analytical models for Ekman layer arrest summarized in BL10 predict a similarly strong decrease of the arrest time scale for large S. A direct comparison with these models is, however, complicated by the fact that they contain, besides S, only one additional parameter (d = CdZ, see above) to describe the arrest time scale, whereas dimensional analysis suggests two (Z and R). Further, for dimensional reasons, the bottom roughness can only appear in the combination z0/V. This, however, is in conflict with the shape of the parameter d, which implicitly depends on z0 via the drag coefficient Cd but exhibits no dependency on V.

The distortion of isopycnals due to the upslope transport of dense water in our simulations generates a continuously increasing pool of APE that reaches its maximum at the time of full Ekman layer arrest. For its quantification, we introduce the nondimensional energy ratio
e31
which compares the APE defined in (19) at full Ekman layer arrest (t = TE) with the amount of energy that has been dissipated into heat during the adjustment period (t < TE).

Figures 10a,b illustrate that Q is never smaller than 10%, but may become several times larger if the slope Burger number is small. For rough bottom (large R) and strong stratification (large Z) values of Q may exceed 70%. This surprising result shows that sloping topography is relatively effective in converting kinetic energy into APE, which provides a remarkable distinction from BBLs over flat topography, in which by far the largest part of the energy is lost to dissipation. The strong increase of Q for decreasing S is easily understood by reconsidering the variability of the arrest time scales discussed above (see Fig. 9). Small values of S were shown to correspond to large TE, which leaves more time for the upslope transport of dense fluid, and thus for the generation of APE.

Fig. 10.
Fig. 10.

Variability of (a),(b) the energy ratio Q and (c),(d) the bulk mixing efficiency Γ as functions of S, Z, and R. Thin contour lines indicate selected values. Only simulations to the right of the thick contour line reached full Ekman arrest (i.e., TE/Tf < 50). Note the logarithmic contour scale in (c) and (d).

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0041.1

In a similar way, we quantify the cumulative efficiency of mixing according to
e32
which compares the total gain of background potential energy due to mixing to the total amount of energy lost to dissipation. This quantity is analogous to the standard mixing efficiency defined in (26), with the difference that it represents an integrated (rather than instantaneous) measure for the mixing efficiency in a turbulent BBL during buoyancy arrest (see related discussion in Smyth et al. 2001).

The dependency of Γ on S and R for the same intermediate value of the frequency ratio used above (Z = 100) is shown in Fig. 10c. The mixing efficiency increases with increasing bottom roughness (increasing R) and increasing slope Burger number but does not exceed Γ = 0.05 even for the highest values of R and S. Similar trends and magnitudes are observed in the dependency of Γ on S and the frequency ratio Z for the intermediate roughness parameter R = 10−4 (Fig. 10d). An interesting exception, however, is the strong increase of Γ for small Z and large S, for which values of up to Γ ≈ 0.2 are reached (Fig. 10d), indicating that BBL mixing may be as efficient as interior mixing in this extreme case. Here, the near-bottom layer behaves like a marginally unstable stratified shear layer (Smyth and Moum 2013) in which stratification, rather than bottom proximity, sets the turbulent length scale. Mixing has been shown to be highly efficient in such cases (e.g., Smyth et al. 2001).

Figures 11a,b show the nondimensional dissipation rate , averaged over the full simulation period until Ekman arrest has occurred. Surprisingly, in spite of the wide parameter range and large differences in flow structure, this scaled quantity is seen to exhibit only a small variability. We use this finding to formulate a simple parameterization for the net energy dissipation in a BBL during Ekman arrest:
e33
where the model constant CE ≈ 1 is seen to represent the time-averaged within a factor of 2 over most of the parameter space (Figs. 11a,b). This provides a convenient way to estimate the energy dissipation for a given along-slope speed V, using TE from Fig. 9.
Fig. 11.
Fig. 11.

As in Fig. 10, but for (a),(b) the nondimensional averaged dissipation rate and (c),(d) the nondimensional averaged mixing rate. Note the logarithmic contour scale in (c) and (d).

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0041.1

The mixing rate is then obtained by multiplying with the mixing efficiency shown in Figs. 10c,d. The nondimensional version of the mixing rate, , averaged over the time scale TE for full Ekman arrest, is displayed in Fig. 11. Average mixing is seen to be strongest on steep slopes (large S) with rough bottom (large R, Fig. 11c) or weak stratification (small Z, Fig. 11d). The former is consistent with the larger dissipation rates over rough bottom whereas the latter may be explained by the combination of efficient mixing and large dissipation rates.

b. Downwelling-favorable flow

The analysis of the downwelling-favorable case (υ > 0) is complicated by the appearance of the BBL oscillations already encountered in section 5. In all cases we investigated, these oscillations started to significantly affect the flow long before Ekman arrest occurred, and they continued to exist without any visible decay even a long time after arrest. The interpretation of the results is more problematic here compared to the upwelling-favorable case because the oscillations affect the energetics of the flow in a possibly artificial way.

For these reasons, we investigated only a small subset of the full parameter space for the downwelling-favorable case to illustrate its basic features, and to discuss the main differences compared to the upwelling-favorable case. The frequency ratio and roughness number were fixed at the intermediate values Z = 100 and R = 10−4, respectively, and only the variability with respect to the slope Burger number S was analyzed. Ekman layer arrest was determined as described in the previous section, however, using the somewhat larger threshold for the drag coefficient (or nondimensional bottom stress). Because of fluctuations of associated with the BBL oscillations close to Ekman arrest, the original threshold of did not yield stable estimates in this case. The practical relevance of this change was found to be small.

Figure 12a illustrates that, for all values of S, oscillations set in long before full Ekman layer arrest is achieved, underlining the complications described above. The decrease of the arrest time scale with increasing slope Burger number is qualitatively similar to the case with upslope Ekman transport (see Fig. 9), but the time scales are about a factor of 4 larger (Fig. 12b). This disparity of the arrest time scales has also been emphasized in previous studies comparing up- and downwelling-favorable flow (e.g., Middleton and Ramsden 1996; BL10). BL10 also provide an analytical expression relating the nondimensional arrest time scale to S and the parameter d = CdZ. With the caveat that a direct mapping between d and our nondimensional parameters is not possible (see above), we find that for Z = 100 and Cd = 0.0029, the model of BL10 and our results show a very similar behavior (besides a constant factor introduced by the different definitions of the arrest time scale, see above).

Fig. 12.
Fig. 12.

Variability of (a) normalized time scales for Ekman layer arrest (full line) and for the appearance of BBL oscillations; (b) ratio of arrest time scales for down- and upwelling favorable flow; (c) nondimensional maximum thicknesses of the BBL, , and the gravitationally unstable layer inside the BBL, . Note that we have chosen Z = 100 and R = 10−4 here. Gray-shaded areas indicate regions of incomplete Ekman layer arrest.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0041.1

While the maximum thickness dBBL of the BBL varies over two orders of magnitude for the range of slope Burger numbers investigated here, a nondimensionalization of the form removes much of this variability (Fig. 12c). A scaling of this form is also consistent with available analytical expressions for the BBL thickness (e.g., Trowbridge and Lentz 1991; BL10), which suggest a decrease of for increasing S that is similar to our results. Interestingly, gravitationally unstable layers are only observed for small S (mild slopes and/or small stratification), but under these conditions their thickness dCL may comprise a substantial fraction of the total thickness of the BBL. The increasing tendency for restratification for increasing S compensates the shear-induced destabilization of the BBL even for relatively small slope Burger numbers (here S < 0.25) to the extent that gravitationally unstable layers do not occur.

Figure 13 illustrates some important differences in the energetics between the up- and downwelling-favorable cases, using the nondimensional quantities derived in the previous sections. To illustrate the uncertainty introduced by the presence of the BBL oscillations, results for the downwelling-favorable case are analyzed for different simulation periods: (i) until full Ekman arrest has occurred and (ii) until oscillations start growing.

Fig. 13.
Fig. 13.

Variability of (a) ratio of APE to dissipated energy at Ekman arrest, (b) cumulative mixing efficiency, (c) integrated nondimensional energy dissipation, and (d) integrated nondimensional mixing rate for up- and downwelling-favorable flow. The meaning of the different line types is explained in (a). Gray-shaded areas, and values of Z and R, are as in Fig. 12.

Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0041.1

The ratio Q [the efficiency of APE generation, see (31)] decreases with increasing Burger number in all cases (Fig. 13a). The ratio Q is smaller in the downwelling case, especially when S is small. Nonetheless, the APE may amount to up to 40% of the dissipated energy. The cumulative mixing efficiency Γ defined in (32) shows a qualitatively similar behavior for both up- and downwelling-favorable flow, but mixing is much less efficient for the latter. Mixing efficiencies are only weakly affected by the BBL oscillations (cf. full and dashed lines), and they are smallest for small S, when the BBL becomes gravitationally unstable. This is analogous to the nonrotating case discussed in Umlauf and Burchard (2011).

The lower cumulative mixing efficiency in the downwelling-favorable case is due largely to the higher cumulative energy dissipation (Fig. 13c), which in turn results from the longer arrest time scales. The cumulative mixing rate shown in Fig. 13d does not follow this trend. BBLs on steeper slopes (larger S) tend to restratify more quickly, which partly compensates for the shorter arrest time scales.

7. Conclusions

One important result of our study is the surprisingly effective conversion of background kinetic energy into APE during buoyancy arrest. Although most of the kinetic energy extracted from the geostrophically balanced alongslope flow is lost to dissipation, the amount of energy stored in the form of APE after full Ekman arrest may be of the same order of magnitude: the ratio of APE creation to dissipation is up to 70% (40%) in the upwelling (downwelling) cases. This energy is, by definition, directly available for conversion into kinetic energy, turbulence, and mixing during the relaxation from buoyancy arrest. The generation of APE is most effective for small Burger numbers, where longer arrest time scales result in larger cross-slope isopycnal excursions, and thus higher APE. While the bottom roughness has a relatively small effect on the generation of APE, the impact of stratification, measured by the nondimensional frequency ratio Z = N/f, cannot be ignored. The largest generation of APE occurs for strong interior stratification, provided that the slopes are small (large Z and small S).

An interesting phenomenon, found in both simulations and field observations (Moum et al. 2004; Holtermann and Umlauf 2012), is the generation of unstable stratification inside the BBL in situations with downslope Ekman transport, resulting in near-bottom convection. Such convective layers were found only for weakly stratified conditions and mild slopes (S < 0.25), and mixing rates and mixing efficiencies for these cases were relatively small.

The upwelling-favorable cases showed generally higher mixing rates and mixing efficiencies compared to their downwelling-favorable counterparts. In these cases, the integrated mixing efficiency, was typically below 5%, with the largest values observed for steep slopes (large S), rough bottom (large R), and strong stratification (large Z). An interesting exception was found for weak interior stratification and very steep slopes (α ≳ 0.1). Under these conditions, the BBL degenerates into a marginally unstable, stratified shear layer with mixing efficiencies up to Γ = 0.2.

These mixing efficiencies, summarized in Figs. 10c,d, may be useful for estimating mixing rates and irreversible buoyancy fluxes from turbulence microstructure measurements. However, it should be kept in mind that most of the energy dissipation in the BBL occurs in a very thin near-bottom layer that is usually not resolved with turbulence microstructure profilers. Assuming, as an example, a logarithmic wall layer for the lowest meter of the water column, it is easy to show that more than 65% of the energy dissipation in this layer occurs in the lowest 0.1 m (assuming a typical bottom roughness of z0 = 10−3 m). The effect of this unresolved near-bottom layer, therefore, has to be accounted for when converting dissipation rates to mixing rates, using the mixing efficiencies found here.

Finally, we note that the energy diagram summarized in Fig. 2 does not involve assumptions about the nature of the forcing terms u and υ, and is, therefore, applicable for any configuration that is consistent with the geometry outlined in section 2. Beyond the sloping Ekman layers investigated here, this also includes a wide range of oscillating BBLs in the ocean and in lakes, and the analysis of idealized numerical simulations of mixing during the near-critical reflection of internal tides (e.g., Chalamalla and Sarkar 2015). For the latter, internal-wave fluxes might have to be taken into account if a finite control volume is considered.

Acknowledgments

LU is grateful for the support by the German Research Foundation (DFG) through Grant UM79/6-1. WDS’s participation was funded by the National Science Foundation under Grant OCE 1355678. Additional support was provided by a Faculty Internationalization Grant from Oregon State University to WDS. JNM’s participation was funded by the Office of Naval Research and the National Science Foundation (1256620). The authors thank the two anonymous reviewers for their insightful comments and suggestions, which helped to significantly improve this manuscript.

APPENDIX

Turbulence Model

a. Turbulence closure model

Here, we briefly review the turbulence closure model used to calculate the turbulent fluxes. A fuller description is given in Umlauf and Burchard (2005). The slope-normal turbulent fluxes appearing on the right-hand sides of (6) are computed from the standard flux-gradient relations:
ea1
where and denote the turbulent diffusivities of momentum and buoyancy, respectively. The latter are assumed to be related to the turbulence kinetic energy, k, and a turbulence length scale, l, according to
ea2
where cμ and are the so-called stability functions. These functions mirror the effects of shear and stratification on the anisotropy of turbulence, and, therefore, constitute one of the most essential components of our turbulence model. Closed polynomial expressions for cμ and can be derived from the full transport equations of the second moments of turbulent quantities under certain equilibrium assumptions. In their review of second-moment modeling, Umlauf and Burchard (2005) discuss how different closure assumptions result in different sets of stability functions, and show that the model of Canuto et al. (2001) yields the physically most consistent results. In this model, used also in the present study, the expressions for cμ and are polynomial functions of the nondimensional time-scale ratios Nk/ε and Mk/ε, where M denotes the total vertical shear. The exact functional forms, and a discussion of the relative advantages of this model with respect to previous formulations, is discussed in Umlauf and Burchard (2005) and Burchard and Bolding (2001).
The length scale l appearing in (A2) is found from the well-known cascading relation lk3/2ε−1 that ties small-scale energy dissipation to the large-scale turbulence parameters k and l (Pope 2000). The dissipation rate is computed from the transport equation:
ea3
where c1c3 are model constants. The production terms P and B defined in (11) can be written as
ea4
using (A1), and assuming that differences between the slope-normal and vertical diffusivities are negligible (α ≪ 1). The transport terms appearing in (10) and (A3) are given by
ea5
where σk and σε are constant Schmidt numbers. These and all other model parameters are summarized in Table A1.
Table A1.

Model parameters appearing in (10) and (A3). The different values for the parameter c3 correspond to stable () and unstable () stratification, respectively.

Table A1.
Umlauf and Burchard (2003) showed that, using the boundary conditions in (9), and assuming negligible stratification, there exists a near-bottom region in which the solutions of (6), (10), and (A3) converge to the well-known law-of-the-wall relations:
ea6
and
ea7
where z0 denotes the bottom roughness, uf is the bottom friction velocity, and κ = 0.42 is the von Kármán constant. To ensure consistency with these solutions, we used lower boundary conditions of the form ∂k/∂z = 0 for the k equation in (10), and for the ε equation in (A3), both of which are obvious limits of (A7) for z = 0. Note that the bottom roughness z0 enters the problem exclusively through the boundary condition for the ε equation.

The numerical procedures for the solution of (6), (10), and (A3) are based on the implementation of the General Ocean Turbulence Model (GOTM) that was used in our study. (The GOTM code and a detailed description of the finite-difference approach are publicly available from www.gotm.net.) Additional details may be found in Umlauf and Burchard (2005), Burchard and Petersen (1999), and Umlauf et al. (2005). All simulations were carried out with a time step of 10 s and 400 equidistant vertical layers. In view of the large variability of the BBL thickness, we adjusted the vertical size of the numerical domain for each simulation, based on predictions of the BBL thickness from the model of BL10. Convergence checks showed that numerical errors were insignificant.

b. Model validation

The turbulence model described above has been tested against a wealth of data from stratified shear flows. Umlauf and Burchard (2005) showed that this model accurately reproduces available data for shear-driven entrainment, an essential requirement for any study focusing on BBL mixing. Burchard and Petersen (1999, see their Fig. 8) demonstrated that the kε model provides a good representation of the Monin–Obukhov theory for stably and unstably stratified boundary layer flows. Particularly relevant for our study is the model performance in stratified shear layers, as these layers model the upper part of the BBL. This class of flows was investigated by Umlauf (2009), who showed that the model almost perfectly reproduces the turbulent diffusivity (and therefore the turbulent buoyancy flux) from a large-eddy simulations (LES) of a stratified shear flow (see their Fig. 5). The convective case was investigated by Umlauf and Burchard (2005). Figure 10b of their study shows that entrainment rates and profiles of the turbulent buoyancy flux are in excellent agreement with LES. A previous study by Burchard and Petersen (1999, see their Figs. 4 and 5) had shown similarly good agreement between the model and laboratory data on free convection.

Finally, in order to study the sensitivity of our results with respect to the choice of the turbulence model, we performed spot checks with the Mellor–Yamada level-2.5 model (Mellor and Yamada 1982) and the kω model described in Umlauf et al. (2003) and Umlauf and Burchard (2003). Similarly to BL10, we find that differences between these models are generally small enough to support all major conclusions in this paper. In particular, the high levels of APE storage at arrest in both upwelling and downwelling are reproduced using the alternative closures, as is the small mixing efficiency (Γ < 0.05). In view of the idealized nature of our study, we therefore believe that our simulations are robust, and provide a credible representation of turbulence and mixing in stratified BBLs.

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